What 3D shapes occur in nature. How efficiently can you pack these shapes together?
This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.
The second in a series of articles on visualising and modelling shapes in the history of astronomy.
Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .
How efficiently can you pack together disks?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
An introduction to bond angle geometry.
Find the point whose sum of distances from the vertices (corners)
of a given triangle is a minimum.
A cube is made from smaller cubes, 5 by 5 by 5, then some of those
cubes are removed. Can you make the specified shapes, and what is
the most and least number of cubes required ?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
A circular plate rolls in contact with the sides of a rectangular
tray. How much of its circumference comes into contact with the
sides of the tray when it rolls around one circuit?
See if you can anticipate successive 'generations' of the two
animals shown here.
This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!
Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Place a red counter in the top left corner of a 4x4 array, which is
covered by 14 other smaller counters, leaving a gap in the bottom
right hand corner (HOME). What is the smallest number of moves. . . .
Glarsynost lives on a planet whose shape is that of a perfect
regular dodecahedron. Can you describe the shortest journey she can
make to ensure that she will see every part of the planet?
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Blue Flibbins are so jealous of their red partners that they will
not leave them on their own with any other bue Flibbin. What is the
quickest way of getting the five pairs of Flibbins safely to. . . .
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
A game for 2 players
On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?
There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being
visible at any one time. Is it possible to reorganise these cubes
so that by dipping the large cube into a pot of paint three times
you. . . .
A bus route has a total duration of 40 minutes. Every 10 minutes,
two buses set out, one from each end. How many buses will one bus
meet on its way from one end to the other end?
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Can you find a rule which connects consecutive triangular numbers?
Can you find a rule which relates triangular numbers to square numbers?
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?
To avoid losing think of another very well known game where the
patterns of play are similar.
Show that all pentagonal numbers are one third of a triangular number.
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?
Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?